Bivariant Intersection Theory

Fulton, William

doi:10.1007/978-1-4612-1700-8_18

Cited by 589 publications

(1,219 citation statements)

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“…This shows that, aside from extending the range for n, the best one can hope for in Theorem 1.4 is a reduction of the lower bound on the Q-order of C to ord Q (C) 10. Moreover, our example shows that it is really the Q-order of C that matters rather than the order, since we could replace C by C + LQ for a linear form L and in this way increase the order of the cubic form.…”

Section: Introductionmentioning

confidence: 99%

Rational Points on Intersections of Cubic and Quadric Hypersurfaces

Browning¹,

Dietmann²,

Heath‐Brown³

2014

J. Inst. Math. Jussieu

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Section: Introductionmentioning

confidence: 99%

Rational Points on Intersections of Cubic and Quadric Hypersurfaces

Browning¹,

Dietmann²,

Heath‐Brown³

2014

J. Inst. Math. Jussieu

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“…This result of Moe and Qviller [30] generalizes a previous result of Eklund, Jost and Peterson [12] which gave an expression for the Segre class of a subscheme of P n in terms of residual sets having a similar structure. For both cases, the residual sets are in the sense of Fulton's residual intersection theorem/formula (Theorem 9.2 and Corollary 9.2.3 of Fulton [16]). …”

Section: Previous Algorithmsmentioning

confidence: 99%

“…the coefficient if α has only one term) of the part of α in the dimension zero Chow group A 0 (V ), see Fulton [16,Definition 1.4] for more details.…”

Section: The Segre Classmentioning

confidence: 99%

Computing characteristic classes of subschemes of smooth toric varieties

Helmer

2017

Journal of Algebra

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Let X Σ be a smooth complete toric variety defined by a fan Σ and let V = V (I) be a subscheme of X Σ defined by an ideal I homogeneous with respect to the grading on the total coordinate ring of X Σ . We show a new expression for the Segre class s(V, X Σ ) in terms of the projective degrees of a rational map specified by the generators of I when each generator corresponds to a numerically effective (nef) divisor. Restricting to the case where X Σ is a smooth projective toric variety and dehomogenizing the total homogeneous coordinate ring of X Σ via a dehomogenizing ideal we also give an expression for the projective degrees of this rational map in terms of the dimension of an explicit quotient ring. Under an additional technical assumption we construct what we call a general dehomogenizing ideal and apply this construction to give effective algorithms to compute the Segre class s(V, X Σ ), the Chern-Schwartz-MacPherson class c SM (V ) and the topological Euler characteristic χ(V ) of V . These algorithms can, in particular, be used for subschemes of any product of projective spaces P n 1 × · · · × P n j or for subschemes of many other projective toric varieties. Running time bounds for several of the algorithms are given and the algorithms are tested on a variety of examples. In all applicable cases our algorithms to compute these characteristic classes are found to offer significantly increased performance over other known algorithms.

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“…It was noticed in [SZ] that the classical Bezout inequality in algebraic geometry [F,Sec. 8.4] together with the Bernstein-Kushnirenko-Khovanskii bound [B, Ku, Kh] produces a new inequality involving mixed volumes of convex bodies:…”

Section: Introductionmentioning

confidence: 99%

Characterization of simplices via the Bezout inequality for mixed volumes

Saroglou

Soprunov

Zvavitch

2016

Proc. Amer. Math. Soc.

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Abstract. We consider the following Bezout inequality for mixed volumes:It was shown previously that the inequality is true for any n -dimensional simplex ∆ and any convex bodies K1, . . . , Kr in R n . It was conjectured that simplices are the only convex bodies for which the inequality holds for arbitrary bodies K1, . . . , Kr in R n . In this paper we prove that this is indeed the case if we assume that ∆ is a convex polytope. Thus the Bezout inequality characterizes simplices in the class of convex n -polytopes. In addition, we show that if a body ∆ satisfies the Bezout inequality for all bodies K1, . . . , Kr then the boundary of ∆ cannot have strict points. In particular, it cannot have points with positive Gaussian curvature.

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Bivariant Intersection Theory

Cited by 589 publications

References 0 publications

Rational Points on Intersections of Cubic and Quadric Hypersurfaces

Rational Points on Intersections of Cubic and Quadric Hypersurfaces

Computing characteristic classes of subschemes of smooth toric varieties

Characterization of simplices via the Bezout inequality for mixed volumes

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